Doubtful, unless Hermione hacks into the system to add the capabilities. Harry could do it if he wanted, but tells me doing it the normal way is easier.
How much you wanna bet she hacks it and adds a web browser anyway? She strikes me as the hacking type.
Author's response
- Lets just say that no one has seen the Ravenclaw Library yet, nor what Harry will do to it 3rd year...
- But no web browser.